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  {\bf Model for Time Perception}
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The inter spike interval $tau \sim$ Exponential($\lambda$), where $1/ \lambda$ denotes the firing rate of a given neuron. Let $t$ be the time for the neuron to fire $n$ spikes, $ t = \sum_{i=1}^N \tau_i$. We have $t \sim \gamma(n, \lambda)$. And $\mu(t) = n\lambda$, and $\sigma(t) = \sqrt{n}\lambda$. Note that when $n = 16$, $\frac{\sigma}{\mu} = 1 / \sqrt{n} = 0.25$. When the number spikes $n$ is fixed, the time it takes to fire a fixed $N = 16$ spikes follows the Weber's law. 

Suppose we have a population of neurons with different firing rates, each of which presents different possible duration of time interval. To remember a time interval $T$, the brain will activate the neuron with firing rate $1/\lambda = N/T$.

In the Markov model, the hidden state for the time perception task is then the true elapsed time $t$, where the observable is the number of spikes $N_t$ at time $t$. In the time discrimination task, the hidden states are then $long$ and $short$ representing the test cue is longer or shorter than the standard cue, respectively.

For a fixed neuron with firing rate $1/\lambda$, we have $P(t_1|N_1,\lambda) \sim \gamma(N_1, \lambda)$ and $P(t_2|N_2, \lambda) \sim \gamma(N_2, \lambda)$. Then $t_1/(t_1+t_2) \sim \beta(N_1, N_2)$. Let $t_2$ be the perceived time the standard duration and $t_1$ be the perceived time up to current time $t$. Then the belief that the current perceived time is longer than the standard duration given the current spike count $N_t$ is
\begin{eqnarray}
  b_t &=& P(t_1 > t_2 | N_t, N_0 = 16) \nonumber \\ 
     &=& P( t_1/(t_1+t_2) > 0.5 | N_t, 16) \nonumber \\
     &=& 1 - betacdf(0.5, N_t, 16)
\end{eqnarray}

The belief that the hidden state is $long$ over time is shown in the following figure:


\includegraphics[scale=0.25]{betacdf.jpg}

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